We make use of the Zeeman's data built in the R "Cusp" package [1]. There are three datasets obtained from three different settings of a Zeeman catastrophe machine. This machine was an architecture made for demonstration of the deterministic cusp catastrophe model: x + y z z3 = 0 with different settings of (x, y). Notice that we changed our coordinate system from (y, α, β) to (x, y, z) here to be consistent to Zeemen's notations.

To demonstrate the proposed approach in Section 3, we utilize the dataset 1 which consists of 150 observations generated from experiments from this Zeeman's machine. The data have three columns of (x, y, z) which are the asymmetry (x) that is orthogonal to the central axis, the bifurcation (y) that is parallel to the central axis and state variable (z) that is the shortest distance from the wheel strap point to the central axis.

4.2 Stochastic Cusp Catastrophe Model Fitting

Similar to the analysis in [1], we fit a series cusp catastrophe models and the best cusp catastrophe model with corresponding equations in equation (4) is as follows:

y = w1Zi , a i = a 0 + a1 X i , b i = b 0 + b1 yi

with estimated parameters of a�O = 0.45, a�1 = 1.15, /3r = 0.99, /3r = -1.48 and

w�1 = 0.90 which are all statistically significant. With this model, the value of the log-

likelihood function is 74.7 and the corresponding value for the linear model is -170.4

which yielded a likelihood-ratio χ2 = 490.3 with degree of freedom of 1 leading to a very small p-value < 0.00001 indicating the cusp model fitted the data better than the linear model. Other model goodness-of-fit statistics, such as R2, AIC and BIC, gives the same conclusion. The Zeeman data (points) and the best fitted response surface of the stochastic cusp catastrophe model are illustrated in Figure 1.

Fig. 1. Zeeman's Data (points) and the fitted cusp catastrophe model (surface)

4.3 Power Calculation and Sample Size Determination

To implement the simulation-based approach in Section 3, we use these estimated parameters with different sample sizes from 10 to 30 by 5, and run the Steps 2 to 6 for 1,000 simulations for each sample size of 10, 15, 20, 25 and 30. The estimated values of statistical power are 0.26, 0.83, 0.96, 0.97 and 0.98, respectively. Figure 2 graphically illustrates the relationship between the sample size and the corresponding statistical power. It can be seen that as sample size increases from 10 to 30, the statistical power increases from 0.26 to 0.98.

In order to get the required sample size for specific statistical power, we can make use of this estimated power curve as illustrated in Figure 2 and back-calculate the

Fig. 2. Power curve to Zeeman's data 1. The arrow-headed line segments indicated the backcalculation of sample size of 15 from power 0.8.

sample size for the specific power. For example in Figure 2, we illustrate this approach for power at 0.8. For power of 0.8, we can back-calculate the sample size based on this estimated power curve to estimate the required sample size which is 15.

4.4 Reverse-Verification

If this novel simulation-based approach is valid, the sample size estimate of 16 described in Section 4.3 would allow approximately 80% chance to detect the underlying cusp catastrophe. Therefore, we took a reverse approach to compute statistical power by repeatedly sample 16 data points from Zeeman's dataset and fitting the stochastic cusp catastrophe model to detect cusp catastrophe. Among 1,000 repeats of the Monte-Carlo simulations with sample size n = 16, we found 851 times (85.1%) significant. This result indicates that the power analysis of the novel simulation-based method is very close to 85%. In another word, the method we proposed is valid.

5 Discussions

In this paper, we proposed a novel Monte-Carlo simulation-based approach to estimate the statistical power for the stochastic cusp catastrophe model. With is approach, we could produce the power curve which depicts the relationship between its statistical power and samples size under different specifications. This power curve can then be used to estimate the sample size required for specified power in design and analysis data from cusp catastrophe model.

We validated this proposed approach by a reverse-verification and demonstrated it is very reasonable. This validation can be further verified by empirically examining Zeeman's data generation. The Zeeman's data were generated for each y with exactly 15 observations for x from -7 to 7 by unit 1 for each y from -2.5 to 2 by 0.5 to produce 150 (= 15×10) observations. Therefore, for each value of y, the 15 data points should be able to generate a cusp catastrophe model.

Acknowledgements. This research was supported in part by three NIH grants from the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD, R01HD075635, PIs: Chen X and Chen D), the National Institute On Drug Abuse (NIDA, R01 DA022730, PI: Chen X) and University of Rochester CTSA award number KL2 TR000095 from the National Center for Advancing Translational Sciences (PI: Lin).

References

1. Grasman, R.P., van der Mass, H.L., Wagenmakers, E.: Fitting the cusp catastrophe in R: A cusp package primer. Journal of Statistical Software 32(8), 1–27 (2009)

2. Thom, R.: Structural stability and morphogenesis. Benjamin-Addison-Wesley, New York (1975)

3. Cobb, L., Ragade, R.K.: Applications of Catastrophe Theory in the Behavioral and Life Sciences. Behavioral Science 23, 291–419 (1978)

4. Cobb, L., Zacks, S.: Applications of Catastrophe Theory for Statistical Modeling in the Biosciences. Journal of the American Statistical Association 80(392), 793–802 (1985)

5. Cobb, L.: An Introduction to cusp surface analysis. Technical report. Aetheling Consultants, Louisville (1998)

6. Hartelman, A.I.: Stochastic catastrophe theory. University of Amsterdam, Amsterdam (1997)

Found a mistake? Please highlight the word and press Shift + Enter